40.3.1 problem 2

Internal problem ID [8608]
Book : ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG, EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section : Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.4. Bessels Equation page 195
Problem number : 2
Date solved : Tuesday, September 30, 2025 at 05:39:44 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {4}{49}\right ) y&=0 \end{align*}

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 36

Order:=6; 
ode:=x^2*diff(diff(y(x),x),x)+x*diff(y(x),x)+(x^2-4/49)*y(x) = 0; 
dsolve(ode,y(x),type='series',x=0);
 

\[ y = \frac {c_2 \,x^{{4}/{7}} \left (1-\frac {7}{36} x^{2}+\frac {49}{4608} x^{4}+\operatorname {O}\left (x^{6}\right )\right )+c_1 \left (1-\frac {7}{20} x^{2}+\frac {49}{1920} x^{4}+\operatorname {O}\left (x^{6}\right )\right )}{x^{{2}/{7}}} \]

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 52

ode=x^2*D[y[x],{x,2}]+x*D[y[x],x]+(x^2-4/49)*y[x]==0; 
ic={}; 
AsymptoticDSolveValue[{ode,ic},y[x],{x,0,5}]
 

\[ y(x)\to c_1 x^{2/7} \left (\frac {49 x^4}{4608}-\frac {7 x^2}{36}+1\right )+\frac {c_2 \left (\frac {49 x^4}{1920}-\frac {7 x^2}{20}+1\right )}{x^{2/7}} \]

✓ Sympy. Time used: 0.316 (sec). Leaf size: 49

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**2*Derivative(y(x), (x, 2)) + x*Derivative(y(x), x) + (x**2 - 4/49)*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics,hint="2nd_power_series_regular",x0=0,n=6)
 

\[ y{\left (x \right )} = C_{2} x^{\frac {2}{7}} \left (\frac {49 x^{4}}{4608} - \frac {7 x^{2}}{36} + 1\right ) + \frac {C_{1} \left (\frac {49 x^{4}}{1920} - \frac {7 x^{2}}{20} + 1\right )}{x^{\frac {2}{7}}} + O\left (x^{6}\right ) \]